﻿488 
  II. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  

  

  from 
  Jupiter's 
  quit 
  is 
  between 
  90° 
  and 
  180° 
  is 
  *224 
  irnvm^r 
  2 
  . 
  

   The 
  number 
  of 
  the 
  comets 
  for 
  which 
  @<r 
  that 
  have 
  inclina- 
  

   tions 
  to 
  the 
  ecliptic 
  less 
  than 
  90° 
  is 
  to 
  the 
  number 
  that 
  have 
  

   inclinations 
  greater 
  than 
  90° 
  as 
  701 
  is 
  to 
  224. 
  Of 
  the 
  839 
  

   comets 
  spoken 
  of 
  in 
  Art, 
  36, 
  203 
  will 
  after 
  perturbation 
  have 
  

   retrograde 
  motions, 
  and 
  636 
  will 
  have 
  direct 
  motions. 
  

  

  40. 
  If 
  to" 
  is 
  less 
  than 
  90° 
  the 
  expression 
  to 
  be 
  integrated 
  in 
  

   order 
  to 
  cover 
  the 
  area 
  SAFD 
  will 
  be 
  

  

  y-»sin 
  «" 
  pi 
  sin 
  \u" 
  n>\ 
  

  

  & 
  ds 
  + 
  / 
  ( 
  O 
  - 
  $') 
  ds+ 
  &"ds. 
  

   4/2 
  — 
  1 
  •'sine/' 
  ^sinw" 
  

  

  4/2- 
  

  

  the 
  corresponding 
  expression 
  

  

  If 
  a>" 
  

  

  is 
  

  

  g 
  

  

  •eater 
  

  

  than 
  90 
  

  

  the 
  

  

  C01 
  

  

  becomes 
  

  

  

  

  

  

  

  

  

  

  

  

  /•2 
  sin-|w" 
  /»2 
  sm-£w 
  y 
  

  

  / 
  & 
  ds— 
  1 
  $" 
  ds. 
  

   " 
  4/2-1 
  ^l 
  

  

  As 
  the 
  value 
  of 
  @ 
  introduces 
  into 
  #' 
  and 
  ®" 
  only 
  one 
  radical 
  

   in 
  s, 
  and 
  that 
  a 
  radical 
  of 
  the 
  second 
  degree, 
  these 
  integrations 
  

   are 
  possible. 
  Finite 
  summation 
  is 
  however 
  more 
  convenient 
  

   Computing 
  the 
  values 
  for 
  each 
  interval 
  of 
  15° 
  we 
  construct 
  the 
  

   following 
  table. 
  The 
  first 
  column 
  indicates 
  the 
  interval 
  in 
  values 
  

   of 
  (o 
  n 
  ; 
  the 
  second 
  column 
  gives 
  that 
  coefficient 
  of 
  ^irnvm^r 
  2 
  that 
  

   must 
  be 
  used 
  to 
  obtain 
  the 
  number 
  of 
  comets 
  which 
  in 
  a 
  unit 
  

   of 
  time 
  will 
  pass 
  perihelion 
  nearer 
  than 
  Jupiter's 
  distance 
  to 
  the 
  

   sun, 
  shall 
  also 
  have 
  their 
  periodic 
  times 
  reduced 
  to 
  be 
  less 
  than 
  

   Jupiter's 
  period, 
  and 
  shall 
  also 
  leave 
  Jupiter's 
  vicinity 
  so 
  that 
  

   the 
  distance 
  between 
  the 
  quits 
  of 
  the 
  two 
  bodies 
  is 
  between 
  the 
  

   two 
  values 
  in 
  column 
  I; 
  the 
  third 
  column 
  indicates 
  the 
  distri- 
  

   bution 
  of 
  the 
  839 
  comets 
  of 
  Art. 
  36 
  through 
  the 
  twelve 
  zones. 
  

  

  

  

  Table 
  III. 
  

  

  

  

  

  Coefficient 
  of 
  

  

  No. 
  out 
  of 
  

  

  Limiting 
  

  

  values 
  of 
  u". 
  

  

  iirnvm°?~~. 
  

  

  839 
  comets. 
  

  

  From 
  

  

  0° 
  to 
  15° 
  

  

  26 
  

  

  6 
  

  

  From 
  

  

  15° 
  to 
  30° 
  

  

  401 
  

  

  91 
  

  

  From 
  

  

  30° 
  to 
  45° 
  

  

  751 
  

  

  J 
  70 
  

  

  From 
  

  

  45° 
  to 
  60° 
  

  

  670 
  

  

  152 
  

  

  From 
  

  

  60° 
  to 
  75° 
  

  

  548 
  

  

  124 
  

  

  From 
  

  

  75° 
  to 
  90° 
  

  

  443 
  

  

  101 
  

  

  From 
  

  

  90° 
  to 
  105° 
  

  

  296 
  

  

  67 
  

  

  From 
  

  

  105° 
  to 
  120° 
  

  

  235 
  

  

  53 
  

  

  From 
  

  

  120° 
  to 
  135° 
  

  

  162 
  

  

  37 
  

  

  From 
  135° 
  to 
  150° 
  

  

  99 
  

  

  23 
  

  

  From 
  

  

  150° 
  to 
  165° 
  

  

  50 
  

  

  11 
  

  

  From 
  

  

  165° 
  to 
  180° 
  

  

  16 
  

  

  4 
  

  

  We 
  see 
  also 
  from 
  the 
  last 
  column 
  of 
  this 
  table 
  that 
  of 
  the 
  

   839 
  comets 
  under 
  consideration 
  267 
  have 
  quits 
  less 
  than 
  45° 
  

  

  